Abstract: A transformation formula under modular substitutions is derived for a very large class of generalized Eisenstein series. The result also gives a transformation formula for generalized Dedekind eta-functions. Various types of Dedekind sums arise, and reciprocity laws are established.

[19]E.
T. Whittaker and G.
N. Watson, A course of modern analysis, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1996. An introduction to
the general theory of infinite processes and of analytic functions; with an
account of the principal transcendental functions; Reprint of the fourth
(1927) edition. MR
1424469

AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and services marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office.